bblee@UniMAP
1
Part III
ERT 216 HEAT & MASS TRANSFER
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7. Convection Mass Transfer Coefficients
8. Mass Transfer Coefficients for various geometries
9. Mass Transfer to Suspensions of Small Particles
Part III
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7. Convection Mass Transfer Coefficients7.3 Mass transfer coefficients for
general case of A and B diffusing and convective flow using film theory
7.4 Mass transfer coefficients under high flux conditions.
8. Mass Transfer Coefficients for various geometries
8.1 Dimensionless numbers used to correlate data
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8.2 Derivation of Mass-transfer coefficients in laminar flow
8.3 Mass transfer for flow inside pipes8.4 Mass transfer for flow outside solid
surfaces9. Mass Transfer to Suspensions of Small
Particles9.1 Introduction9.2 Equations for mass transfer to small
particles
bblee@UniMAP5
Assuming a simplified film theory where the mass transfer is assumed to occur through a thin film next to the wall of thickness f and by molecular diffusion.
The experimental value of kc for dilute solutions is used to determine the film thickness f :
f
AB'
c
Dk
bblee@UniMAP6
Rewriting:
The convective term is
Rearranging and integrating:
BAAAAB
A NNxdz
dxDcN
BAA NNx
2
10
1 A
A
f x
xBAAA
Az
zAB NNxN
dxdz
cD
bblee@UniMAP7
1
2
ABAA
ABAA'
c
BA
AA
xNNN
xNNNlnck
NN
NN
bblee@UniMAP8
In section 7.3, it is assumed that the film thickness is unaffected by high fluxes and bulk or convective flow (diffusion-induced convection). In the case of A diffusing through
stagnant, nondiffusing B where diffusion-induced convection is present.For the flux NA at the surface z = 0
where xA=xA1,01
0
AA
z
AABA Nx
dz
dxcDN
Not average value
bblee@UniMAP9
Defining a mass-transfer coefficient in terms of the diffusion flux,
Solving for NA,
At low concentrations & fluxes, kc0
approaches kc for no bulk flow:
21
0
0
AAc
z
AAB xxck
dz
dxcD
1
21
0
1 A
AAcA
x
xxckN
21 AA
'
cA xxckN
bblee@UniMAP10
A coefficient kc may be defined without regard to convective flow:
Combining:
The relationship between kc0 or kc for high flux and kc for low fluid will be derived using the film theory.
21 AAcA xxckN
cAc kxk 10 1
bblee@UniMAP11
Based on film theory, for transferring of A by molecular diffusion and convection flow with B being stagnant & nondiffusing.Setting NB=0,
For the film theory,
2121 AAcAA
BM
'
cA cckcc
x
kN
'
x
x
BM
'
c
c
k
k
xk
k 1
bblee@UniMAP12
Combining,
'
x
x
BM
A
'
c
c
k
k
x
x
k
k 010 1
EXAMPLE 7.2-2
bblee@UniMAP13
EXAMPLE 7.2-2
bblee@UniMAP14
8.1 Dimensionless numbers used to correlate data
Schmidt number (NSc) The ratio of the shear component for
diffusivity (/) to the diffusivity for mass transfer DAB.
It physically relates the relative thickness of the hydrodynamic layer and mass transfer boundary layer.
AB
ScD
N Viscosity
Density
bblee@UniMAP15
Schmidt number (NSc): the ratio of momentum diffusivity to
molecular diffusivity.
ABABAB
ScD
D
D
N
bblee@UniMAP16
Reynolds number (NRe) the most important dimensionless no. it indicates degree of turbulence.
LvNRe
Diameter (Dp) for a sphere or Diameter for a pipe or Length for a flat plate.
Mass average velocity (in a pipe) or Superficial velocity (v) in the empty cross section of a packed bed or Interstitial velocity (v/); =void fraction of bed.
bblee@UniMAP17
Sherwood number (Nsh): It is also known as mass transfer
Nusselt number. It represents the ratio of convective to
diffusive mass transport.
AB
'
x
AB
BMc
AB
'
cshD
L
c
k
D
Lyk
D
LkN
bblee@UniMAP18
Take the case of gas phase mass transfer for flow past a sphere, 1cm in diameter, at low partial pressure of the solute.
s/m
ms/m~y
D
DkN BM
AB
cSh 25
22
10
1010= 10
sm
sm
D
N
AB
Sc 25
25
10
10= 1
bblee@UniMAP19
For liquid phase mass transport in a similar geometry,
s/m
ms/m~y
D
DkN BM
AB
cSh 29
25
10
1010= 100
sm
sm
D
N
AB
Sc 29
26
10
10= 1000
bblee@UniMAP20
8.2.1 IntroductionWhen a fluid is flowing in laminar flow and
mass transfer by molecular diffusion is occurring.The equations are very similar to those for
heat transfer by conduction in laminar flow(But not always true).In mass transfer several components may
be diffusing, the flux of mass perpendicular to the direction of the flow must be small (not distort the laminar velocity profile).
bblee@UniMAP21
The mass transfer coefficients for laminar flow for certain geometries (e.g. flow past a cylinder or in a packed bed) are difficult to be described using mathematical approach. Experimental mass-transfer
coefficients are often obtained and correlated.
A simplified theoretical derivation will be given for two cases of laminar flow.
bblee@UniMAP22
8.2.2 Mass transfer in laminar flow in a tube
In the case of mass transfer from a tubewall to a fluid inside in laminar flow, the wall is made of solid benzoic acid which is dissolving in water.This is similar to heat transfer from a
wall to the flowing fluid where natural convection is negligible.
bblee@UniMAP23
For fully developed flow, the parabolic velocity derived as:
For steady state diffusion in a cylinder, a mass balance can be made on a differential element where the rate in by convection plus diffusion equals the rate out radiallyby diffusion:
2
2
12
1
R
rv
R
rvv
av
maxxVelocity in the x direction at the distance r from the center.
bblee@UniMAP24
2
2
2
21
x
c
r
c
r
c
rD
x
cv AAAAB
Ax
02
2
x
cA
If the diffusion in the x direction is
negligible compared to that by convection,
bblee@UniMAP 25
Mass transfer of solute A into a laminar falling film, which is important in wetted wall columns, in developing theories to explain mass transfer in stagnant pockets of fluids, and in turbulent mass transfer.
Figure 7.3-1: Diffusion of solute A in a laminar falling film: (a) velocity profile & concentration profile.
bblee@UniMAP26
Figure 7.3-1: Diffusion of solute A in a laminar falling film: (b) small element for mass balance.
bblee@UniMAP27
The solute A in the gas is absorbed at the interface and then diffuses a distance into the liquid so that it has not penetrated the whole distance x= at the wall.At steady state the inlet concentration
cA=0.The concentration profile of cA at a point
z distance from the inlet is shown in Fig 7.3-1a.A mass balance on the element is shown in
Fig. 7.3-1b.
bblee@UniMAP28
We obtain:
To determine the local molar flux at the surface x=0 at position z from the top entrance,
2
2
x
cD
z
cv AAB
Az
z
vDc
x
cDzN maxABA
x
AABxAx 0
00
bblee@UniMAP29
The total moles of A transferred per second to the liquid over the entire length z=0 to z=L, where the vertical surface is unit width:
dzN.LNL
AxlxA
0
011
dzz
vDc
/
/
maxAB
L
A 21
21
0
0
11
L
vDc.L maxABA
41 0
bblee@UniMAP30
The rate of mass transfer is proportional to and
The basis for the penetration theory in turbulent mass transfer where pockets of liquid are exposed to unsteady-state diffusion (penetration) for short contact times.
L
max
tv
LTime of exposure of the liquid to the solute A in the gas.
50.
ABD501 .Lt
bblee@UniMAP31
8.3.1 Mass transfer for laminar flow inside pipes.
When a liquid or gas is flowing inside a pipe & NRe (Dv/) is below 2000, laminar flow occurs.
Figure 7.3-2: Data for
diffusion in a fluid in
streamline flow inside a
pipe.
bblee@UniMAP32
The dimensionless coordinates:or
The velocity profile is assumed fully developed to parabolic form at the entrance.
LD
W
AB
4LDNN ScReFlow (kg/s)
Length of mass transfer section (m)
bblee@UniMAP33
For liquids that have small values of DABData follow the parabolic flow lineFor W/DAB L > 400:
32
0
0 55
/
ABAAi
AA
LD
W.
cc
cc
bblee@UniMAP34
For turbulent flow when NRe (Dv/) is above 2100 for gases or liquids flowing inside a pipe,
The equation holds for NSc of 0.6 to 3000.Note: The NSc for gases is in the range 0.5-
3 and for liquids is above 100 in general.
AB
BMc
AB
'
cShD
D
P
k
D
DkN
330830
0230
.
AB
.
ShD
D.N
bblee@UniMAP35
When a gas is flowing inside the core of a wetted-wall tower, the same correlations that are used for mass transfer of a gas in laminar or turbulent flow in a pipe are applicable. The equations can be used for NRe up to
about 1200.
u
N
avz
Re
44
bblee@UniMAP36
EXAMPLE 7.3-1
bblee@UniMAP37
EXAMPLE 7.3-1
bblee@UniMAP38
The mass transfer and vaporization of liquids from a plate or flat surface to a flowing stream is of interest in the dryingof inorganic & biological materials, in the evaporation of solvents from paints, for plates in wind tunnels & in flow channels in chemical process equipment.When the fluid flows past a plate in a free
stream in an open space the boundary layer is not fully developed.
bblee@UniMAP39
8.4.1 Mass transfer in flow parallel to flat plates
For gases or evaporation of liquids in the gas phase & for the laminar region of NRe,L=Lv/ < 15 000, the data can be represented within 25% by:
Writing in terms of Sherwood number:
506640 .LRe,D N.J
31506640 /Sc.
LRe,Sh
AB
'
c NN.ND
LkLength of the plate in the direction of flow
bblee@UniMAP40
For gases & NRe,L of 15 000 30 000, the data are represented within 30% by JD=JH=f/2:
Experimental data for liquids are correlated within about 40% for a NRe,Lof 600 50 000 by:
200360 .LRe,D N.J
50990 .LRe,D N.J
bblee@UniMAP41
For flow past single spheres and for very low NRe=Dpv/ (v average velocity in the empty test section before the sphere), The NSc = kcDp/DAB should approach 2.0.
The mass transfer coefficient kc which is kc for a dilute solution:
2121
2AAcAA
p
ABA cckcc
D
DN
p
AB'
cD
Dk
2
Diameter
bblee@UniMAP42
Rearranging,
Note: Natural convection effects could increase kc
For gases, for NSc of 0.6 2.7 and NRe of 1 48 000:
02.ND
DkSh
AB
p
'
c
3153055202 /Sc.
ReSh NN.N
bblee@UniMAP43
For liquids, NRe of 2 to about 2000,
For liquids, NRe of 2000 17 000,
315009502 /Sc.
ReSh NN.N
3162034702 /Sc.
ReSh NN.N
bblee@UniMAP44
EXAMPLE 7.3-3
bblee@UniMAP45
EXAMPLE 7.3-3
bblee@UniMAP46
EXAMPLE 7.3-3
bblee@UniMAP47
EXAMPLE 7.3-3
bblee@UniMAP48
Mass transfer to and from packed beds occurs often in processing operations, including drying operations, adsorption or desorption of gases or liquids by solid particles such as charcoal, & mass transfer of gases & liquids to catalyst particles.By using a packed bed a large amount of
mass-transfer area can be contained in a relatively small volume.
bblee@UniMAP49
The void fraction () in a bed is m3 volume void space divided by the m3 total volume of void space plus solid.The values range from 0.3 to 0.5.Because of flow channeling, nonuniform
packing, accurate experimental data are difficult to obtain and data from different investigators can deviate considerably.
bblee@UniMAP50
For NRe range of 10 10 000 for gases in a packed bed of spheres, the recommended correlation with an average deviation of about 20% and a maximum of about 50%:
4069045480 .ReHD N
.JJ
'vDN
p
Re
Diameter of the spheres
Superficial mass average velocity in the empty tube
without packing.
bblee@UniMAP51
For NRe range of 0.0016 55 & NSc range of 165 70 000,
For liquids, NRe range of 55 - 1500 & NScrange of 165 10 690,
For liquids, NRe range of 10 - 1500,
32091 /ReD N
.J
310250 .ReD N
.J
4069045480 .ReD N
.J
bblee@UniMAP52
For fluidized beds of spheres, for gases and liquids, NRe range of 10 4000,
For liquids in a fluidized bed, NRe range of 1 10,
72010681 .ReD N.J
4069045480 .ReD N
.J
Nota: If packed beds of solids other than spheres are used, approximate correction factors can be used. Particle diameter = surface area of the solid particle.
bblee@UniMAP53
The total flux in a packed bed:1. JD is obtained 2. kc (m/s) is determined from JD3. Total volume Vb (m
3) of the bed [voids + solids]
4. Total external surface area A (m2) of the solids for mass transfer:
pD
a
16Surface area (m2) / total volume of bed (m3) [when the solids are spheres]
baVA
bblee@UniMAP54
5. To calculate the mass transfer rate, the log mean driving force at the inlet & outlet of the bed:
2
1
21
AAi
AAi
AAiAAicA
cc
ccln
ccccAkAN
Concentration at the surface of the solid
(kg mol / m3)
Inlet bulk fluid concentration (kg mol / m3)
Outlet bulk fluid
concentration (kg mol / m3)
bblee@UniMAP55
6. The material-balance equation on the bulk stream:
The equations can be used for a fluid flowing in a pipe or past a flat pipe, where A = the pipe wall area or plate area.
12 AAA ccVAN
Volumetric flow rate of fluid entering (m3/s)
bblee@UniMAP56
9.1 Introduction Mass transfer from/to small suspended
particles in an agitated solution occurs in a number of process applications. In liquid-phase hydrogenation, hydrogen
diffuses from gas bubbles, through an organic liquid, & then to small suspended catalyst particles.
In fermentation, oxygen diffuses from small gas bubbles, through the aqueous medium, & then to small suspended microorganisms.
bblee@UniMAP57
For liquid-solid suspension, increased agitation over & above that necessary to freely suspend very small particles has very little effect on the mass transfer coefficient, kL to the particle.When the particles in a mixing vessel are
just completely suspended, turbulence forces balance those due to gravity.
bblee@UniMAP58
With very small particles (m), which is the size of many microorganisms in fermentations and some catalyst particles, their size is smaller than eddies, which are about 100 m or so in size. increased agitation will have little
effect on mass transfer except at very high agitation.
bblee@UniMAP59
For a gas-liquid-solid dispersion (e.g. fermentation), increased agitation increases the number of gas bubbles and hence the interfacial area. The mass-transfer coefficients from
the gas bubble to the liquid and from the liquid to the solid are relatively unaffected.
bblee@UniMAP60
9.2.1 Mass transfer to small particles < 0.6mm
Equations for predicting mass transfer to small particles in suspension have been developed which cover 3 size ranges of particles:1) < 0.60mm2) >2.5 mm3) 0.60 - 2.5mm
bblee@UniMAP61
To predict mass-transfer coefficients from small gas bubbles (e.g. oxygen or air to the liquid phase or from the liquid phase to the surface of small catalyst particles, microorganisms, other solids, liquid drops):
31
2
323102
/
c
c/
Sc
p
AB'
L
gN.
D
Dk
Diffusivity
of the solute A in solution
(m2/s)
Diameter of the gas bubble / solid particle
(m)
Viscosity of the
solution (kg/m.s)
9.80665 m/s2
Density of continuous
phase (kg/m3)
Density of the gas or
solid particles (kg/m3)
bblee@UniMAP62
In aerated mixing vessels, the mass-transfer coefficients are essentially independent of the power input.
31
2
323102
/
c
c/
Sc
p
AB'
L
gN.
D
Dk
Molecular diffusion
Due to free fall / rise of the sphere due to gravitational
forces
bblee@UniMAP63
EXAMPLE 7.4-1
bblee@UniMAP64
EXAMPLE 7.4-1
bblee@UniMAP65
EXAMPLE 7.4-1
bblee@UniMAP66
As the power is increased, the bubble size decreases & the mass transfer coefficient continues to follow the above equation.The dispersions include those in which
the solid particles are just completely suspension of these small particles results in only small increase in kL.
bblee@UniMAP67
For large gas bubbles or liquid drops > 2.5mm,
Large gas bubbles are produced when pure liquid are aerated in mixing vessels & sieve plate columns.kL or kL is independent of the bubble size
and is constant for a given values of kLabout 3 4 times larger than that of small particles.
31
2
50420
/
c
c.
Sc
'
L
gN.k
bblee@UniMAP68
kL is essentially independent of agitation intensity in an agitated vessel and gas velocity in a sieve-tray tower.
bblee@UniMAP69
In mass transfer in the transition region between small & large bubbles in the size range of 0.6 - 2.5 mm, the mass transfer coefficient can be approximated by assuming that it increases linearly with bubble diameter.
bblee@UniMAP70
In the preceding 3 regions, the density difference between phases is sufficiently large to cause the force of gravity to primarily determine the mass-transfer coefficient.This includes solids just completely
suspended in mixing vessels.
bblee@UniMAP71
When agitation power is increased beyond that needed for suspension of solid or liquid particles & the turbulence forces becomes larger than the gravitational forces. Small increases in kL:
41
2
32 130
/
c
c/
Sc
'
L
VP.Nk
Power input per unit volume
bblee@UniMAP72
In the case of gas-liquid dispersions it is quite impractical for agitation systems to exceed gravitational forces.The experimental data are complicated by
the fact that very small particles are easily suspended, & if their size is on the order of the smallest eddies, the mass transfer coefficient will remain constant until a large increase in power input is added above that required for suspension.
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